Problem Set
Multiple Choice Test
Chapter 01.01
Introduction to Numerical Methods
COMPLETE SOLUTION SET
1. Solving an engineering problem requires four steps. In order of sequence the four
steps are
(A) formulate, model, solve, implement
(B) formulate, solve, model, implement
(C) formulate, model, implement, solve
(D) model, formulate, implement, solve
Solution
The correct answer is (A).
The four steps of solving an engineering problem are:
1) Formulate the problem (same as describing the problem)
2) Mathematically model the problem
3) Solve the mathematical model
4) Implement the results in engineering practice
2. One of the roots of the equation x 3 − 3 x 2 + x − 3 = 0 is
(A) -1
(B) 1
(C) 3
(D) 3
Solution
The correct answer is (D).
x 3 − 3x 2 + x − 3 = 0
x 2 ( x − 3) + 1( x − 3) = 0
( x 2 + 1)( x − 3) = 0
Therefore, x = 3 is a solution to the above equation.
3. The solution to the set of equations
25a + b + c = 25
64a + 8b + c = 71
144a + 12b + c = 155
most nearly is (a, b, c ) =
(A) (1,1,1)
(B) (1,-1,1)
(C) (1,1,-1)
(D) does not have a unique solution.
Solution
The correct answer is (C).
25a + b + c = 25 (1)
64a + 8b + c = 71 (2)
144a + 12b + c = 155 (3)
Subtracting Equation (1) from Equation (2) gives
39a + 7b = 46 (4)
Subtracting Equation (1) from Equation (3) gives
119a + 11b = 130 (5)
From Equation (4),
46 − 7b
a= (6)
39
Substituting the value of a from Equation (6) in Equation (5) gives
46 − 7b
119 + 11b = 130
39
140.36 − 21.359 + 11b = 130
− 10.358b = −10.36
− 10.36
b=
− 10.359
= 1.0001
From Equation (4),
46 − 7(1.0001)
a=
39
= 0.99998
From Equation (1),
c = 25 − 25a − b
= 25 − 25(0.99998) − 1.0001
= −0.99960
So
(a, b, c ) = (0.99998,1.0001,−0.99960)
≈ (1,1,−1)
π
4
4. The exact integral of ∫ 2 cos 2 xdx is most nearly
0
(A) -1.000
(B) 1.000
(C) 0.000
(D) 2.000
Solution
The correct answer is (B).
π
4
∫ 2 cos 2 xdx
0
π
sin(2 x) 4
= 2
2 0
π
= [sin(2 x )]04
π
= sin 2 − sin(2(0) )
4
π
= sin − sin(0)
2
= 1− 0
=1
dy
5. The value of (1.0) , given y = 2 sin (3x ) most nearly is
dx
(A) -5.9399
(B) -1.980
(C) 0.31402
(D) 5.9918
Solution
The correct answer is (A).
y = 2 sin (3 x )
dy
= 2(3 cos(3 x) )
dx
= 6 cos(3 x)
dy
(1.0) = 6 cos(3(1.0) ) (Remember the argument of trig functions is radians)
dx
= 6(− 0.98999)
= −5.9399
6. The form of the exact solution of the ordinary differential equation
dy
2 + 3 y = 5e − x , y (0 ) = 5 is
dx
−1.5 x
(A) Ae + Be x
(B) Ae −1.5 x + Be − x
(C) Ae1.5 x + Be − x
(D) Ae −1.5 x + Bxe − x
Solution
The correct answer is (B).
dy
2 + 3 y = 5e − x , y (0 ) = 5
dx
The characteristic equation for the homogeneous part of the solution is
2m1 + 3m 0 = 0
2m + 3 = 0
m = −1.5
The homogeneous part of the solution hence is
y H = Ae −1.5 x
The particular part of the solution is
y P = Be − x
So the form of the solution to the ordinary differential equation is
y = yH + yP
= Ae −1.5 x + Be − x
